Theorem crnggrpd 20467

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20466	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20462	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Grpcgrp 19137  CRingccrg 20453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-ov 7421  df-ring 20454  df-cring 20455

This theorem is used by:  fermltlchr  21828  selvvvval  22444  selvadd  22445  psdmul  22480  psd1  22481  psdpw  22484  ply1chr  22617  ply1fermltlchr  22623  elrgspnsubrunlem1  33801  elrgspnsubrunlem2  33802  erlbr2d  33818  rlocaddval  33823  rloccring  33825  rloc0g  33826  rlocf1  33828  fracerl  33861  gsumind  33899  dflringlem2  34020  ressply1evls1  34090  evl1deg1  34101  evl1deg2  34102  evl1deg3  34103  ply1dg1rt  34105  vr1nz  34118  mplasclco  34141  psrmonprod  34177  mplmonprod  34179  esplyfvn  34202  irngss  34312  extdgfialglem1  34317  irredminply  34341  algextdeglem4  34345  algextdeglem5  34346  aks6d1c1p3  43140  aks6d1c2lem4  43157  aks6d1c6lem2  43201  aks5lem2  43217  evlsbagval  43594  evlselv  43597  prjcrv0  43649